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Properties

Label 2-294-1.1-c5-0-28
Degree 22
Conductor 294294
Sign −1-1
Analytic cond. 47.152847.1528
Root an. cond. 6.866796.86679
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 9·3-s + 16·4-s + 54·5-s − 36·6-s − 64·8-s + 81·9-s − 216·10-s + 216·11-s + 144·12-s − 998·13-s + 486·15-s + 256·16-s − 1.30e3·17-s − 324·18-s − 884·19-s + 864·20-s − 864·22-s − 2.26e3·23-s − 576·24-s − 209·25-s + 3.99e3·26-s + 729·27-s − 1.48e3·29-s − 1.94e3·30-s − 8.36e3·31-s − 1.02e3·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.965·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.683·10-s + 0.538·11-s + 0.288·12-s − 1.63·13-s + 0.557·15-s + 1/4·16-s − 1.09·17-s − 0.235·18-s − 0.561·19-s + 0.482·20-s − 0.380·22-s − 0.893·23-s − 0.204·24-s − 0.0668·25-s + 1.15·26-s + 0.192·27-s − 0.327·29-s − 0.394·30-s − 1.56·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 294, ( :5/2), −1)(2,\ 294,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+p2T 1 + p^{2} T
3 1−p2T 1 - p^{2} T
7 1 1
good5 1−54T+p5T2 1 - 54 T + p^{5} T^{2}
11 1−216T+p5T2 1 - 216 T + p^{5} T^{2}
13 1+998T+p5T2 1 + 998 T + p^{5} T^{2}
17 1+1302T+p5T2 1 + 1302 T + p^{5} T^{2}
19 1+884T+p5T2 1 + 884 T + p^{5} T^{2}
23 1+2268T+p5T2 1 + 2268 T + p^{5} T^{2}
29 1+1482T+p5T2 1 + 1482 T + p^{5} T^{2}
31 1+8360T+p5T2 1 + 8360 T + p^{5} T^{2}
37 1+4714T+p5T2 1 + 4714 T + p^{5} T^{2}
41 1−9786T+p5T2 1 - 9786 T + p^{5} T^{2}
43 1−452pT+p5T2 1 - 452 p T + p^{5} T^{2}
47 1+22200T+p5T2 1 + 22200 T + p^{5} T^{2}
53 1−26790T+p5T2 1 - 26790 T + p^{5} T^{2}
59 1+28092T+p5T2 1 + 28092 T + p^{5} T^{2}
61 1−38866T+p5T2 1 - 38866 T + p^{5} T^{2}
67 1−23948T+p5T2 1 - 23948 T + p^{5} T^{2}
71 1+20628T+p5T2 1 + 20628 T + p^{5} T^{2}
73 1+290T+p5T2 1 + 290 T + p^{5} T^{2}
79 1+99544T+p5T2 1 + 99544 T + p^{5} T^{2}
83 1+19308T+p5T2 1 + 19308 T + p^{5} T^{2}
89 1+36390T+p5T2 1 + 36390 T + p^{5} T^{2}
97 1−79078T+p5T2 1 - 79078 T + p^{5} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.13138878422012828275125896465, −9.479238417332253732506700943464, −8.805146359968703072457581298369, −7.59686932792248376289971633177, −6.74287202410047817896836804514, −5.59745284526236769814277142920, −4.15819202142693109125843893692, −2.48258085168926064275249611604, −1.81189726127868307563972767213, 0, 1.81189726127868307563972767213, 2.48258085168926064275249611604, 4.15819202142693109125843893692, 5.59745284526236769814277142920, 6.74287202410047817896836804514, 7.59686932792248376289971633177, 8.805146359968703072457581298369, 9.479238417332253732506700943464, 10.13138878422012828275125896465

Graph of the ZZ-function along the critical line